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For which value of n below is a perfect square

(2^8)+(2^11)+(2^n)

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Re: Perfect Square [#permalink ]
21 May 2011, 05:28

2^8(1 + 2^3) + 2^n

= 2^8 * (9 + (2)^n-8)

= 9 + (2)^n-8 = 25 then its a perfect square

=> (2)^n-8 = 16

=> (2)^n-8 = 2^4

=> n = 12

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Re: Perfect Square [#permalink ]
21 May 2011, 05:29

subhashghosh wrote:

2^8(1 + 2^3) + 2^n = 2^8 * (9 + (2)^n-8) = 9 + (2)^n-8 = 25 then its a perfect square => (2)^n-8 = 16 => (2)^n-8 = 2^4 => n = 12

yup thats the answer

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Re: Perfect Square [#permalink ]
21 May 2011, 05:55

@subhashghosh can you please explain me this part?

subhashghosh wrote:

= 2^8 * (9 + (2)^n-8) = 9 + (2)^n-8 = 25 then its a perfect square

Thanks.

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Re: Perfect Square [#permalink ]
21 May 2011, 06:02

hussi9 wrote:

For which value of n below is a perfect square (2^8)+(2^11)+(2^n)

(2^8)+(2^11)+(2^n)

=(2^(4*2))+(2. 2^4 .2^6 )+

(2^n) To complete the square n ==12 i.e 6*2

(2^2)^2 + +(2. 2^4 .2^6 ) +

(2^6)^2 _________________

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Re: Perfect Square [#permalink ]
21 May 2011, 07:23

wish answer options were given

would have done answer fitting.

2^8[1+2^3+ 2^(n-8)] means

9+2^(n-8) = perfect square meaning 2^(n-8) = perfect square

16,64 are possible values.

n=12 fits.

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Re: Perfect Square [#permalink ]
21 May 2011, 07:32

@jamifahad, consider this :

2^8 * (9 + (2)^n-8) is a product of 2^8 (perfect square) and (9 + (2)^n-8)

So we want (9 + (2)^n-8) to be a perfect square.

Now 2^any number is even while 9 is odd

So we want 9 + even # to be a perfect square, and that will be and odd #.

So check for instance 9 + powers of two which can be a perfect square.

After a few calculations by checking powers of 2 in increasing order, you can see that 2^4 = 16, and 16 + 9 = 25, a perfect square.

Please let me know if you have any other query.

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Re: Perfect Square [#permalink ]
21 May 2011, 08:24

@subhash got it thanks.

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Re: Perfect Square [#permalink ]
21 May 2011, 08:39

(2^8)+(2^11)+(2^n)

must be equal to (a+b)^2 form

=>(2^8)+(2^11)+(2^n) = a^2 + b^2 + 2ab

2^11 looks like a 2ab form . if we consider 2^8 = a^2 => a = 2^4

=> b must be 2^11 / (2*2^4) => b = 6

also n = 2b = 12

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Re: Perfect Square [#permalink ]
21 May 2011, 13:39

2^8+2^11+2^n taking 2^8 common we have (2^8)(1+2^3+2^(n-8)) = (2^8)(9+2^(n-8)) we know that 2^8 is a perfect square . so 9+2^(n-8) has to be perfect square minimum power of 2 that can be added to 9 to make it a perfect square is 4.( 9+16 = 25) => n -8 =4 Hence n=12.

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Re: Perfect Square [#permalink ]
18 Aug 2012, 01:30

subhashghosh wrote:

2^8(1 + 2^3) + 2^n = 2^8 * (9 + (2)^n-8) = 9 + (2)^n-8 = 25 then its a perfect square => (2)^n-8 = 16 => (2)^n-8 = 2^4 => n = 12

just a small query, we are assuming that 2^n > 2^8 , which enables us to take 2^8 as a common factor

and we get 2^8 ( 2^ n-8 )

but what if 2^n = 4 then n = 2 in which case we cannot take 2^8 as a common factor .

any thoughts as to how we can easily assume 2^n > 2^8 ?

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Re: Perfect Square [#permalink ]
18 Aug 2012, 02:22
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hussi9 wrote:

For which value of n below is a perfect square (2^8)+(2^11)+(2^n)

Use the formula for

(a+b)^2=a^2+2ab+b^2. Since

2^8+2^{11}=(2^4)^2+2*2^4*2^6 , we need an extra term, that of

(2^6)^2=2^{12} to complete the expression to a perfect square,

(2^4+2^6)^2. So,

n should be

12 .

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Re: Perfect Square [#permalink ]
18 Aug 2012, 17:51

EvaJager wrote:

hussi9 wrote:

For which value of n below is a perfect square (2^8)+(2^11)+(2^n)

Use the formula for

(a+b)^2=a^2+2ab+b^2. Since

2^8+2^{11}=(2^4)^2+2*2^4*2^6 , we need an extra term, that of

(2^6)^2=2^{12} to complete the expression to a perfect square,

(2^4+2^6)^2. So,

n should be

12 .

Thank you eva I think this approach is more justified, like it better

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Re: Perfect Square [#permalink ]
10 Sep 2012, 14:11

gurpreetsingh wrote:

(2^8)+(2^11)+(2^n) must be equal to (a+b)^2 form =>(2^8)+(2^11)+(2^n) = a^2 + b^2 + 2ab 2^11 looks like a 2ab form . if we consider 2^8 = a^2 => a = 2^4 => b must be 2^11 / (2*2^4) => b = 6 also n = 2b = 12

Your approach is absolutely right, except some ambiguity. Let me correct it.

Where from did you get n=2b ? Also, b is not equal to 6. Correct expression is

b=2^6 As

2^n = b^2 So,

2^n= (2^6)^2=2^{12} Therefore, n=12

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Re: Perfect Square
[#permalink ]
10 Sep 2012, 14:11